Question

Monitoring phone calls to a toll-free number. A largefood-products company receives about 100,000 phone callsa year from consumers on its toll-free number. A computermonitors and records how many rings it takes for an operatorto answer, how much time each caller spends “on hold,” andother data. However, the reliability of the monitoring systemhas been called into question by the operators and their labour unions. As a check on the computer system, approximatelyhow many calls should be manually monitored during thenext year to estimate the true mean time that callers spend onhold to within 3 seconds with 95% confidence? Answer thisquestion for the following values of the standard deviation ofwaiting times (in seconds): 10, 20, and 30.

Short answer

For a standard deviation of 10

43 calls should be manually monitored during the next year to estimate the true mean time that callers spend on hold to within 3 seconds with 95% confidence

For a standard deviation of 20

171 calls should be manually monitored during the next year to estimate the true mean time that callers spend on hold to within 3 seconds with 95% confidence

For a standard deviation of 30

384 calls should be manually monitored during the next year to estimate the true mean time that callers spend on hold to within 3 seconds with 95% confidence

Step-by-step solution

Given information

Answer this question for the following values of the standard deviation of waiting times (in seconds): 10, 20, and 30.

Finding the sample size

Here the standard error is 3

The critical value for a 95% confidence interval is

$z_{\alpha /2}=z_{0.05/2}=z_{0.025}=1.96$

The value of the standard deviation is 10

$\begin{array}{c}SE=z_{\alpha /2}\frac{\sigma }{\sqrt{n}}\\ n={z^2}_{\alpha /2}\frac{{\sigma }^2}{S{E}^2}\\ n={1.96}^2\times \frac{{10}^2}{3^2}\\ n=42.68444\\ n\simeq 43\end{array}$

Therefore,43 calls should be manually monitored during the next year to estimate the true mean time that callers spend on hold to within 3 seconds with 95% confidence

The value of the standard deviation is 20

$\begin{array}{c}SE=z_{\alpha /2}\frac{\sigma }{\sqrt{n}}\\ n={z^2}_{\alpha /2}\frac{{\sigma }^2}{S{E}^2}\\ n={1.96}^2\times \frac{{20}^2}{3^2}\\ n=170.7378\\ n\simeq 171\end{array}$

Therefore,171 calls should be manually monitored during the next year to estimate the true mean time that callers spend on hold to within 3 seconds with 95% confidence

The value of the standard deviation is 30

$\begin{array}{c}SE=z_{\alpha /2}\frac{\sigma }{\sqrt{n}}\\ n={z^2}_{\alpha /2}\frac{{\sigma }^2}{S{E}^2}\\ n={1.96}^2\times \frac{{30}^2}{3^2}\\ n=384.16\\ n\simeq 385\end{array}$

Therefore,385 calls should be manually monitored during the next year to estimate the true mean time that callers spend on hold to within 3 seconds with 95% confidence